Thermalization of dense hadronic matter in Au + Au collisions at energies available at the Facility for Antiproton and Ion Research

The conditions of local thermodynamic equilibrium of baryons (nonstrange, strange) and mesons (strange) are presented for central Au + Au collisions at Facility for Antiproton and Ion Research (FAIR) energies using the microscopic transport model UrQMD. The net particle density, longitudinal-to-transverse pressure anisotropy, and inverse slope parameters of the energy spectra of nonstrange and strange hadrons are calculated inside a cell in the central region within rapidity window $\left|y\right|<1.0$ at different time steps after the collisions. We observed that the strangeness content is dominated by baryons at all energies; however, contributions from mesons become significant at higher energies. The time scale obtained from local pressure (momentum) isotropization and thermalization of energy spectra are nearly equal and found to decrease with increase in laboratory energy. The equilibrium thermodynamic properties of the system are obtained with a statistical thermal model. The time evolution of the entropy densities at FAIR energies are found to be very similar to the ideal hydrodynamic behavior at top Relativistic Heavy Ion Collider (RHIC) energy.

Figure 1

(Upper panel) Time evolution of net density of (a) nonstrange baryons (${\rho }_B^{NS}$), (b) strange baryons (${\rho }_B^S$); (lower panel) (c) kaons (${\rho }_M^S$) and (d) net strange baryon to kaon ratio (${\rho }_B^S/{\rho }_M^S$) inside the central cell for Au + Au collisions ($b=2$ fm) at the laboratory energies $10A$, $20A$, $30A$, and $40A$ GeV. The error bars are statistical only.

Figure 2

Time evolution of longitudinal-to-transverse pressure ratio ($P_L/P_T$) of nonstrange baryons, strange baryons, and kaons inside the central cell for Au + Au collisions ($b=2$ fm) at the laboratory energies (a) $10A$, (b) $20A$, (c) $30A$, and (d) $40A$ GeV. The error bars are statistical only.

Figure 3

Time evolution of the inverse slope parameter ($T_{\mathrm{slope}}$) of the energy spectra of proton and $\mathrm{\Lambda }$ inside the central cell for Au+Au collisions ($b=2$ fm) at the laboratory energies (a) $10A$, (b) $20A$, (c) $30A$, and (d) $40A$ GeV. The error bars are statistical only.

Figure 4

The scaling behavior of inverse slope parameter ($T_{\mathrm{slope}}$) of the energy spectra of $\mathrm{\Lambda }$ inside the central cell for Au+Au collisions ($b=2$ fm) at (a) $20A$ and (b) $40A$ GeV laboratory energies. The black dotted line denotes the scaling due to longitudinal expansion and the red dashed line denotes scaling due to three-dimensional expansion. The error bars are statistical only.

Figure 5

(Upper panel) The ${\chi }^2$ per degrees of freedom (${\chi }^2/ndf$) at different times for the Tsallis (filled symbols) and Maxwell-Boltzmann (open symbols) distribution which are fitted to the energy spectra of proton and $\mathrm{\Lambda }$ in the central cell for Au +Au collisions ($b=2$ fm) at $E_{\mathrm{lab}}=30A$ GeV. (Lower panel) The thermalization time obtained in this work (blue square) is compared with the local thermalization time $t_{\mathrm{start}}$ (red circle) used in the hybrid (UrQMD + hydrodynamics) model [31]. The error bars are considered to be systematic.

Figure 6

Time evolution of average number of collisions ($\langle N_{\mathrm{coll}}\left(t\right)\rangle$) suffered by (a) protons and neutrons, (b) $\mathrm{\Lambda }$ and $\mathrm{\Sigma }$ baryons, and (c) kaons for Au + Au collisions ($b=2$ fm) at laboratory energies $10A$ and $30A$ GeV.

Figure 7

The evolution of temperature ($T$) and baryochemical potential (${\mu }_B$) in the central cell for Au + Au collisions ($b=2$ fm) at $E_{\mathrm{lab}}=10A$ GeV and $30A$ GeV. The solid (blue) line denotes the chemical freeze-out and the dashed (red) line denotes thermal freeze-out boundary in relativistic heavy ion collisions [36, 37]. The dotted circle denotes the expected region probed by the CBM experiment ($\sqrt{s_{NN}}=4\text{–}10$ GeV) at FAIR.

Figure 8

The time evolution of entropy density of baryonic matter inside the central cell for Au +Au collisions (b = 2 fm) at $E_{\mathrm{lab}}=10A$ GeV and $30A$ GeV. The dashed line denotes the parametrization of ideal hydrodynamic evolution of entropy density in the central region for Au +Au collisions (b = 0 fm) at $\sqrt{s_{NN}}=200$ GeV [44].